The Disjoint Paths problem has been also studied extensively from the viewpoint of parameterized algorithms. Here, the goal is to design an algorithm
which runs in \(f(k)\cdot n^{O(1)}\)  time for a function \(f\) , where \(k\)  is the number of terminal pairs.
The seminal work of Roberson and Seymour [1]}
showed that this problem can be solved in \(f(k)\cdot n^3\)  time for a computable function \(f\) .
Although this algorithm is powerful, the original proof of Roberson and Seymour requires the full power of the Graph Minor Thoery.
For this reason, Kawarabayashi and Wollan [2]} presented a simpler proof of correctness of the algorithm in [1]}.
Later, this algorithm was improved by Kawarabayashi et al. [4]}
to run in \(h(k)\cdot n^2\)  time for a computable function \(h\) .
However, the dependence on \(k\)  is still huge both in [4]}, [1]}. Moreover, the explicit bound on the dependence on \(k\)  is not known.
